We determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

中文翻译：

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长期预测确定性的一致性强度

我们确定长度投影游戏的确定性一致性强度ω2. 我们的主要定理是$\Pi _{n + 1}^1 $- 长度游戏的确定性ω2意味着存在一个集合论模型ω + n伍丁红衣主教。第一步，我们证明这个假设意味着存在可数的实数集一种这样米n(一种)，典型的内部模型n伍丁红衣主教一种, 满足$$A = R$$和确定性公理。然后我们讨论如何获得一个模型ω + nWoodin cardinal from this. 我们还展示了如何调整证明来研究长度游戏确定性的一致性强度ω2有回报$^RR\Pi _1^1 $或与σ- 投射收益。